Tempo Ramps for Phasing Effects and Polytemporal Music

In 2008 I started playing with the construction of tempo ramps used to generate phasing effects, inspired by the compositions of Steve Reich. This experimentation lead to the creation of tempo ramps used in the performance of polytemporal music where the different tempi had no mathematical relationship to each other... which I termed "disassociated tempo."

Phasing Tempo Ramps

A Phasing Tempo Ramp is my term for describing the increase in tempo one player must achieve to end up precisely one beat (or any other predetermined value established by the composer) ahead of those players who stay at the original tempo, when all players reach measure X. In other words, a phasing tempo ramp can last for any number of measures, over which one or more players accelerate to a precise tempo before decelerating to the original tempo to join the other players. And at the end of that ramp, the accelerating player(s) will be precisely one beat (or any fraction or multiple of the beat) ahead of those who did not accelerate.

Let me say that again. I wanted to know the maximum tempo the accelerating player(s) must achieve over a given number of measures to end up precisely one beat (or subdivision/multiple of that beat) ahead of the other players. I also wanted to determine how comfortably that accelerating part could be played.

I settled on two basic shapes of Phasing Tempo Ramps, each of which can be used in a variety of ways.

The Drop Tempo Ramp (Phasing Music)

The Drop Tempo Ramp ramp (shown above) has a steady incline as it accelerates to a specified tempo over a given number of measures, at which point it suddenly drops down to join the original tempo. The ramp can be of any length. The shorter the duration, the faster the acceleration. Conversely, the longer the duration, the slower the acceleration and the greater the chance of creating interference patterns during performance. In the example shown, the Drop Ramp is 8 measures in length, lasting from the start of measure 25 to the start of measure 33. The starting tempo at measure 25 is 97 BPM. The longer the ramp takes to meet the specified tempo (in this case 99.99 BPM), the slower the acceleration rate will be and the more interference patterns (and maybe some actual phasing) will occur.

By adjusting the math, the length of the ramp can be any number of measures. The more measures during which the acceleration occurs, the more interference patterns are likely to occur as the time separating the parts is slowly stretched from unison to some predetermined interval.

The Drop Tempo Ramp is most effective over short intervals of time, but the phasing effect is not as pronounced. I found myself using the Drop Tempo Ramp for dramatic effect at the end of musical segment or form (e.g. AABA), or to adjust the lag (or advancement) of individual parts in preparation for another effect, and in the development of nested ramps (to be discussed later). The only limitation I discovered was the somewhat jarring effect on the performer when asked to accelerate rapidly over a short period of time. Using a Drop Tempo Ramp for just 4 measures will definitely be felt by the performer, while the acceleration caused by using a Drop Tempo Ramp spread over 24 measures is almost imperceptible by the performer ...assuming the advancement at the end of the ramp is just one beat for both cases.

The Triangle Tempo Ramp (Phasing Music)

The image below shows a 12 measure Triangle Tempo Ramp that starts at the beginning of measure 64 with a tempo of 120 BPM and continues to the end of measure 69 (which is also the start of measure 70). The Triangle Tempo Ramp being shown accelerates for just 5 measures. At the end of measure 69, the accelerating part will have reached a tempo of 121.67 BPM (which can seen in the list on the extreme right of the screen,) at which point it starts to decelerate over the next 7 measures to join those parts which maintained the original tempo of 120 BPM. At the point of unification, the accelerated part will be precisely 1/8th note ahead of the other players. In reality, the point of maximum acceleration can be set for any point within the duration of that ramp. For example, if the ramp was 19 measures in length, the maximum tempo could be set to occur at the end of the 11th measure, to be followed by 8 measures of deceleration before joining the original tempo. It's all in the math and the desired musical effect.

The use of a Triangle Tempo Ramp has the advantage of smoothing out the deceleration portion of the ramp, thereby providing a gradual unification of a common tempo. It tends to heighten the sense of awareness of the phasing effect and gives the impression of control and "deliberateness" to the generation of interference patterns. Of the two, the Triangle Tempo Ramp is easier to perform. The acceleration and deceleration can be so gradual that performers at rehearsals are unaware that they were playing a "ramping" part. As such, I find myself using this shape more often than the Drop Tempo Ramp.

As a composer, I wanted to discover the "playability" of the different ramps. I also wanted to know which shapes produced the most pleasing effect. What could the ear follow, since there was bound to be a sense of chaos generated during the phasing period. Was that chaos a pleasant experience? In Reich's case, the "phasing effect" was a new atheistic introduced to the musical audience, and I wanted to explore that effect in as many ways as possible.

Different shapes of ramps can be applied to the same part (instrument), as shown in the image below. Please note that the shorter the length of time (in measures) the ramp takes to reach its intended peak, the steeper the ramp is going to be, and the greater the focus of the performer must also be on following that tempo map.

Stacked and Nested Tempo Ramps

Tempo ramps can also be nested. For example, one performer can have a 12 measure ramp starting at measure 20, while another performer can have an 8 measure ramp starting at measure 24. There is no limitation to the degree of nesting that can be woven into a single composition, but my experiments to date indicate that nesting is best used for polytemporal compositions which have no derivative or mathematical relationships in the tempi between the different parts. In other words, nested tempo ramps used for phasing techniques are very difficult for the ear to follow.

The image on the left displays all of the ramps throughout an entire score, with nested ramps occurring between the 2nd violin and the viola. Two of the players in the quartet, the 1st violin and the cello, have no tempo ramps. If you were to look at a notated score for this composition, you would see that that bar lines do not line up due to the fact that some instruments are accelerating while others are not. This discrepancy requires adjustments in the scores of certain players, which I explain in another section titled Engraving Problems.

Ramps for Dissociated Tempi and Time Signatures

The ramps on the left come from an experiment with dissociated tempi. The effects generated (after a few opening measures to establish sync) come from parts whose tempi are not mathematically related in any obvious way. Hence the term "dissociated." Nor is the tempo in one section derived from the value of a note in a previous section. In other words, one half note in a previous passage does not equal 1 quarter note in the next section; nor does it equal a doted quarter note/double-doted eighth note/doted eighth note inside a triplet, one beat of a septuplet, or any of the other associated "tuplet variations." The time signature is not some weird 23/2, or a time signature where the half note in one measure now equals a doted quarter note in another. None of these techniques are in play.

In this exercise, only instrument #1 sustains the base tempo of 120 BPM throughout. At the start of measure 7, instruments #1, #2, and #3 start a rhythm broken into three complementing patterns (still at 120 BPM.)

Instrument #4 is in 3/4 while the other instruments are in 4/4. Instrument #4 enters at measure 9 with a tempo of 95 BPM and sustains that tempo until measure 57, at which point it jumps to the original tempo of 120 BPM. (Two measures of rest precede this jump in tempo and makes it easier to perform, as demonstrated in the conducted network video.

Instrument #2 starts a Triangle Phasing Ramp at measure 10 and another at measure 40, and each ramp returns to the base tempo of 120 BPM. The maximum acceleration of the first Phasing Ramp is 121.25 BPM and occurs at measure 15. Although the slopes of the two phasing ramps are slightly different in length, they both reach a maximum acceleration rate of 121.25 BPM. The maximum acceleration rate of the second Phasing Ramp occurs at measure 44. In both ramps, instrument #2 returns to the base tempo of 120 BPM. However, instrument #2 will have advanced a predetermined fraction of the beat compared to instrument #1 after each ramp.

Instrument #3 starts a Floating Ramp at measure 30 which does not return to the base tempo of 120 BPM. Therefore, this Floating instrument continues to advance (in terms of beat placement-offset compared to the base tempo held by instrument #1) without actually going through another tempo ramp.

The ramps are so gentle and slow that most instrumentalists will be unaware that their tempi are shifting. As far as audience reaction, if a strong melody is assigned to instrument #4, that melody will become the focus of the casual observer. The first time you practice this exercise, players #1, #2, and #3 will think performer #4 can't count. At that point you can switch parts and watch the light bulbs turn on. This exercise will likely be more fun to play than to hear. The casual observer will hear organized chaos that, at times, evolves into a different pattern of organized chaos.

I never actually finished this experimental piece. I intended to insert a neutralizing tempo ramp into the parts #2, #3 and #4, so that all parts return to the base tempo of 120. Instrument #4 might have switched to 4/4 at the very end.

I have the synchronized conducting video for this exercise and will provide a link on this page once I have the time to find and upload the video. You can then make up your own rhythms and melodies, follow the tempo ramps, and see if your ear can comprehend what's going on in this mixture of dissociated tempi and meters.

A Phased Parsing of Terms

This is a good point to discuss the term "phasing" as it applies to polytemporal music. I spent several years as a recording studio engineer, working on staff at various recording facilities and eventually owning my own studio. Based on my experience and understanding of acoustics and electronics, the use of the term "phasing" in this musical context is a bit of a misnomer. A more accurate term to describe what is most likely to be heard in the application of musical phasing techniques is not phasing, but the creation of "interference patterns." Actual "phasing," in both an electronic and acoustic context, induces a canceling - or sometimes an enhancing-- of frequencies (usually overtones) when two sources producing identical rhythms, pitches and tenor, arrive at our ears (or a microphone) separated by very small differences in time. A common problem with phasing also occurs when two or more microphones are placed fairly close together and capture the sounds from the same instrument. Phasing becomes obvious in a control room where the sound appears in electronic form, where the output of different microphones are mixed together.

Those live concerts involving pop music that have multiple guitar amps on stage with several microphones picking up the same sounds encounter real phasing problems.

Phasing, as a constructive or destructive process, is difficult to predict due to countless variations (electronic and acoustic) that are not under the control of the composer or performer. The acoustics of the venue (especially the stage) play a role in generating (or eliminating) this effect. And in a concert venue involving acoustical instruments, where microphones and amplification are kept at a minimum, acoustic phasing is difficult to predict and achieve. To deliberately produce phasing by using the acoustical properties of the venue, where the sound is emanating from acoustical instruments, the instruments themselves have to be as closely tuned as possible and placed in relatively close proximity on the stage. When identical note patterns are produced by at least two instruments, and where the tempo of one player (or one group of players) begins to accelerate very, very slowly, thereby separating the performance of identical musical patterns in a deliberate, controlled and slow manner (we're talking about differences in microseconds), actual phasing can be produced. But it is not easily accomplished. What you are more likely to hear is a "stuttering" effect where the common patterns are offset by a short interval of time, i.e. the generation of interference patterns.

For me, the performance of Reich's "Piano Phase" on YouTube by Rob Kovacs is one of the best demonstrations of a performance which encourages the production of phasing effects. The two pianos are placed as close together as possible, meaning the acoustics of the venue should be applied equally to both pianos. Both pianos were tuned to each other, and since Kovacs played both pianos, the expectation of matching velocity on the same notes was high.

The YouTube image was not clear enough to identify the placement of any microphones. It is possible there were microphones high in the ceiling providing amplification of the ambient sounds of the stage, but those kinds of microphones do not promote the appearance of phasing artifacts. The tops of both pianos had been removed, but I don't see the placement of a pair of close-proximity microphones just above the strings. If microphones are present, the output of the microphones would have been fed to the house mixer, and if any phasing was being induced by those microphones, it would have been corrected electronically in the control room before being fed to the house PA system. If true phasing were to occur due to the design of the composition, this arrangement should have encouraged the appearance of that phenomenon

Reich's initial introduction to phasing took place while playing two identical lengths of the same recorded material on audio tape using two different tape decks. Due to inherent properties in the construction of analog audio tape decks, the speed of the tape decks tended to vary ever so slightly over a period of time. When both decks are playing the exact same tape loops over and over, but at slightly different speeds, an audible effect known as "phasing" occurs.

But reproducing that kind of precision in a live venue isn't easily predictable. What is more likely to happen is that the repetition of the same sounds (pitch, rhythm and amplitude) coming from two different sources (musicians), arriving at a microphone (or our ears) at very short differences in time. If you place two marimbas on stage and have them play identical notes and rhythms, you likely won't hear any phasing. Even if you ask one of the players to slowly accelerate you won't hear the "hollowing," "swishing" artifacts of phasing. What is more likely to be experienced is a "stuttering" or echoing effect, especially when identical rhythms are performed and repeated (looped). In a live setting, there are just too many variables at work to make real phasing predictable.

The bouncing of the same sounds around the room from two different sources is unlikely to cause real phasing because that bouncing itself causes frequencies from each source to arrive at the microphone (or ear) at different times. On the other hand, deliberately creating interference patterns is relatively easy to achieve, as Reich has demonstrated in several compositions. And interference patterns are even easier to achieve with long, controlled, tempo ramps, using the polytemporal synchronization systems I describe on this website.

The phenomenon of phasing is well understood by recording engineers, and various controls are built into mixing consoles and microphones to compensate for this unwanted effect. Why? Because real phasing results in a drop of volume when the identical sound from two different sources arrives at the same microphone at slightly different times. When phasing happens, there is a "hollowing" effect, a cancelling of the frequencies...a canceling of the amplitude of the waveshapes, electronically speaking. The sound becomes "hollow."

When the acoustical and electronic precautions described are observed and the overtones do not cancel each other, the effect that occurs is more accurately described as the generation of "interference patterns," which can still be an interesting artifact to manipulate. In my mind, generating "interference patterns" better describes the separation of identical rhythms as they drift apart and then re-combine to form different rhythmical structures. If one takes a broad view of Reich's compositional technique, one can substitute the term "permutation" for "phasing" as the combination of two instruments (or groups) playing identical rhythms form new patterns once they are off-set into different permutation of the original structure.

However, for the sake of immediate comprehension and to stay abreast of the jargon, I will acquiesce and refer to this phenomenon as "phasing," as the term is now too well established, and as my membership in the Society of Motion Pictures and Television Engineers (SMPTE) has long since lapsed. Besides. "Interference Pattern Music" just doesn't have the same panache.